Question

If the length of shadow of a pole is equal to the height of the pole, then the angle of elevation of the sun is: -
(a) \[{{30}^{\circ }}\]
(b) \[{{75}^{\circ }}\]
(c) \[{{60}^{\circ }}\]
(d) \[{{45}^{\circ }}\]

Answer 1

Hint: Construct a rough diagram of the given situation and assume angle ‘\[\theta \]’ as the angle of elevation. Assume AB as the pole with ‘A’ as its top and ‘B’ as its bottom. The diagram will look like a right angle triangle with point ‘C’ as the shadow of point ‘A’ and BC will be the length of shadow. Use the formula: - \[\tan \theta =\dfrac{p}{b}\], where p = perpendicular and b = base. Substitute the values of p and b to find the value of \[\tan \theta \] and hence find the value of \[\theta \].

Complete step by step answer:
Let us draw a rough diagram for the given conditions of the question.

We have considered AB as the pole of height ‘h’ with ‘A’ as its top and ‘B’ as its bottom. The sun is present such that the shadow of point A is formed at point C. The angle of elevation of the sun is assumed to be ‘\[\theta \]’.
Clearly, we can see that ABC is a right angle triangle at B.
Now, in right angle triangle ABC, we have,
AB = length of pole = h
BC = length of shadow of pole = h
\[\angle ACB=\theta \] = angle of elevation of the sun.
Therefore, using the relation: - \[\tan \theta =\dfrac{p}{b}\], where p = perpendicular and b = base, we get,
\[\begin{align}
  & \Rightarrow \tan \theta =\dfrac{AB}{BC} \\
 & \Rightarrow \tan \theta =\dfrac{h}{h} \\
 & \Rightarrow \tan \theta =1 \\
\end{align}\]
We know that, \[\tan {{45}^{\circ }}=1\].
\[\Rightarrow \tan \theta =\tan {{45}^{\circ }}\]
Removing tan function from both sides, we get,
\[\Rightarrow \theta ={{45}^{\circ }}\]

So, the correct answer is “Option D”.

Note: One may note that, in the right angle triangle ABC, we have used \[\tan \theta \] and not \[\sin \theta \] or \[\cos \theta \]. This is because we have been provided with the length of base and perpendicular of the triangle and not the hypotenuse. So, if we will use \[\sin \theta \] or \[\cos \theta \] then it will only elongate the process because then we have to find the length of the hypotenuse.